The Defining Ideal of a Set of Points in Multi‐Projective Space

The defining ideal IX of a set of points X in ℙn1×...×ℙn1 is investigated with a special emphasis on the case when X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, the degrees of the generators of the associated ideal IX are determined. ν(IX) denotes the minimal number of generators of IX, and this description of the degrees is used to construct a function υ(s; n1,…,nk) with the property that ν(IX)⩾ υ(s; n1,…, nk) always holds for s points in generic position in ℙn1×...×ℙn1. When k = 1, υ(s;n1) equals the expected value for ν(IX) as predicted by the ideal generation conjecture. If k ⩾ 2, it is shown that there are cases with ν(IX) > υ(s; n1, …, nk). However, computational evidence suggests that in many cases ν(IX) = υ(s; n1, …, nk).